+818 Let c be a real number, and consider the system of quadratic equations
y = x^2 - 9x + c,
y = 8x^2 - 13x.
For which values of c does this system have:
(a) Exactly one real solution (x,y)?
(b) More than one real solution?
(c) No real solutions?
Solutions to the quadratics are (x,y) pairs.
+36 Hint 1: Use the quadratic formula, most notably, the discrimant.
Hint 2: Part a) is a single number, the rest are inequalities.
Here's the answer for part (a):
If you put the 2 equations together, you will get x2−9x+c=8x2−13x, from which you will get 7x2−4x−c=0. In order for their to be only 1 solution, the discriminant must be equal to 0. The discrimant of any quadratic is b2−4ac (It is the part in the square root of the quadratic formula, which is the formula that pops up whenever you press the LaTeX button on your screen.)
b2−4ac=0, so
16+28c=0, from which you can solve
c=−1628=−47
